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How Can Metcalfe's Law Be Updated for Web 2.0?

Metcalfe's Law is More Misunderstood Than Wrong

"Metcalfe's Law is Wrong," contended Bob Briscoe, Andrew Odlyzko, and Benjamin Tilly recently in a much-discussed IEEE Spectrum article, in which they wrote: "Of all the popular ideas of the Internet boom, one of the most dangerously influential was Metcalfe's Law." Sim Simeonov disagrees.

The industry is at it again – trying to figure out what to make of Metcalfe’s Law. This time it’s IEEE Spectrum with a controversially titled “Metcalfe’s Law is Wrong”. The main thrust of the argument is that the value of a network grows O(nlogn) as opposed to O(n2). Unfortunately, the authors’ O(nlogn) suggestion is no more accurate or insightful than the original proposal.

There are three issues to consider:

  • The difference between what Bob Metcalfe claimed and what ended up becoming Metcalfe’s Law
  • The units of measurement
  • What happens with large networks

The typical statement of the law is “the value of a network increases proportionately with the square of the number of its users.” That’s what you’ll find at the Wikipedia link above. It happens to not be what Bob Metcalfe claimed in the first place. These days I work with Bob at Polaris Venture Partners. I have seen a copy of the original (circa 1980) transparency that Bob created to communicate his idea. IEEE Spectrum has a good reproduction, shown here.

The original Metcalfe's Law graph

The unit of measurement along the X-axis is “compatibly communicating devices”, not users. The credit for the “users” formulation goes to George Gilder who wrote about Metcalfe’s Law in Forbes ASAP on September 13, 1993. However, Gilder’s article talks about machines and not users. Anyway, both the “users” and “machines” formulations miss the subtlety imposed by the “compatibly communicating” qualifier, which is the key to understanding the concept.

Bob, who invented Ethernet, was addressing small LANs where machines are visible to one another and share services such as discovery, email, etc. He recalls that his goal was to have companies install networks with at least three nodes. Now, that’s a far cry from the Internet, which is huge, where most machines cannot see one another and/or have nothing to communicate about… So, if you’re talking about a smallish network where indeed nodes are “compatibly communicating”, I’d argue that the original suggestion holds pretty well.

The authors of the IEEE article take the “users” formulation and suggest that the value of a network should grow on the order of O(nlogn) as opposed to O(n2). Are they correct? It depends. Is their proposal a meaningful improvement on the original idea? No.

To justify the logn factor, the authors apply Zipf’s Law to large networks. Again, the issue I have is with the unit of measurement. Zipf’s Law applies to homogeneous populations (the original research was on natural language). You can apply it to books, movies and songs. It’s meaningless to apply it to the population of books, movies and songs put together or, for that matter, to the Internet, which is perhaps the most heterogeneous collection of nodes, people, communities, interests, etc. one can point to. For the same reason, you cannot apply it to MySpace, which is a group of sub-communities hosted on the same online community infrastructure (OCI), or to the Cingular / AT&T Wireless merger.

The main point of Metcalfe’s Law is that the value of networks exhibits super-linear growth. If you measure the size of networks in users, the value definitely does not grow O(n2) but I’m not sure O(nlogn) is a significantly better approximation, especially for large networks. A better approximation of value would be something along the lines of O(SumC(O(mclogmc))), where C is the set of homogeneous sub-networks/communities and mc is the size of the particular sub-community/network. Since the same user can be a member of multiple social networks, and since |C| is a function of N (there are more communities in larger networks), it’s not clear what the total value will end up being. That’s a Long Tail argument if you want one…

Very large networks pose a further problem. Size introduces friction and complicates connectivity, discovery, identity management, trust provisioning, etc. Does this mean that at some point the value of a network starts going down (as another good illustration from the IEEE article shows)? It depends on infrastructure. Clients and servers play different roles in networks. (For more on this in the context of Metcalfe’s Law, see Integration is the Killer App, an article I wrote for XML Journal in 2003, having spent less time thinking about the problem ;-) ). P2P sharing, search engines and portals, anti-spam tools and federated identity management schemes are just but a few examples of the myriad of technologies that have all come about to address scaling problems on the Internet. MySpace and LinkedIn have very different rules of engagement and policing schemes. These communities will grow and increase in value very differently. That’s another argument for the value of a network aggregating across a myriad of sub-networks.

Bottom line, the article attacks Metcalfe’s Law but fails to propose a meaningful alternative.

More Stories By Simeon Simeonov

Simeon Simeonov is CEO of FastIgnite, where he invests in and advises startups. He was chief architect or CTO at companies such as Allaire, Macromedia, Better Advertising and Thing Labs. He blogs at blog.simeonov.com, tweets as @simeons and lives in the Greater Boston area with his wife, son and an adopted dog named Tye.

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Most Recent Comments
wsanders 08/13/06 08:41:43 AM EDT

I sort of see the key insight of Briscoe, Odlyzko, and Tilly that, if you are going to pull a function out of your ass, it makes more sense if the differential of the function flattens out rather than slopes linearly upwards forever, because there is ultimately a decreasing value of each connection as the number of connections increases.

So they were correct to pull a log function of of their ass, but they could have just as easily pulled out n*ln(n) or some other base. They made no attempt to "calibrate" the model.

A good insight is this quotation:

"If Metcalfe's Law were true, it would create overwhelming incentives for all networks relying on the same technology to merge, or at least to interconnect. These incentives would make isolated networks hard to explain. Consider two networks, each with n members. By Metcalfe's Law, each one's value is on the order of n 2, so the total value of both of these separate networks is roughly 2n 2. But suppose these two networks merge. Then we will effectively have a single network with 2n members, which, by Metcalfe's Law, will be worth (2n)2 or 4n 2--twice as much as the combined value of the two separate networks.

Surely it would require a singularly obtuse management, to say nothing of stunningly inefficient financial markets, to fail to seize this obvious opportunity to double total network value by simply combining the two."

Inflating these "synergies" was exactly what led to the Bombing Off of the Bubble.

MountainLogic 08/13/06 08:38:03 AM EDT

This all came about because Metcalfe was trying to make a case for networking (e.g., ethernet).

Back then the ethernet cards he was selling were expensive. The decision maker would go, "gee, if it cost $x to network two people why can't Bob just walk down the hall to Jan's office?" If X is greater then the cost of Bob "walking down the hall" (or snail mailing or flying...) then there is no busines case for installing a network. More to the point:
If the node cost, x, is $100 and there are 100 users, n, then the cost for the network is $10,000.

If the single user business value, v, of the network is $10 for one user then the ROI for different valuation methods is:

Linear: vn = $1,000 -- no business case, don't even think about it

Metcalf's Law: (n(n-1)=2)v = 49,500 -- winner

Metcalf's Law as misused by dot-bombers: N^2 * V = 100,000 -- "Proves" selling frozen mud on the net is a winner

As restated by the authors: n long (n) * v = 2000 -- no business case, but better than a flat linear

There really are two problems here. The scaling formula and setting the business value. If you set the business value for a single connection greater than the cost of the network then it is a no brainer, but back when Metcalfe was pushing networking that was a hard case to make.

Scott Allen 08/13/06 08:20:00 AM EDT

Metcalfe's Law has always been understood to mean "the theoretical potential value", since a very large number of the links between nodes will never be made.

This same idea can be applied to Reed's Law, which states that the potential value of social networks, i.e., those with self-organizing groups, grows as 2^N (actually, 2^N - N - 1). In reality, of course, this never even comes close, because real groups don't usually organize like that ("let's make a sub-group within our group that excludes one person", etc.).

Also, virtual groups, in particular, don't tend to break off into ad hoc sub-groups if they don't have the tools to do so, e.g., the participants in one thread don't typically say "Let's go create a new group to discuss this." It's theoretically possible, but only occasionally occurs. Of course, the participants in a particular thread can be considered a sort of ad hoc sub-group, in which case you might achieve something closer to the potential, but still...

Interesting stuff - thanks for the link.

Jon Rubin 08/13/06 08:05:56 AM EDT

Metcalfe's Law said that the value of a network, x, as it increases in network members, n, is described by the equation x=n^2.

In other words, it said the value of a network was proportional to the square of the network's users.

Instead, the IEEE article declares it should be x=n"log(n). My suspicion - and this has no basis in anything and I haven't even graphed it and my total knowledge of information theory is that it was started by a guy named Claude - is that Euler has to be involved somewhere and maybe x=n"ln(n) would be more correct.

Peter O'Kelly 08/13/06 08:03:17 AM EDT

insightful analysis

Johannes Ernst 08/13/06 07:58:19 AM EDT

Of course Metcalfe's Law is overstated! If I have a fax machine and somebody in central China, who I have never heard of and will never interact with, buys another fax machine, the value of the network will not grow proportionally to N (say, hundreds of millions: the number of fax machines in existence today) but by some much smaller number such as a couple of hundred at the maximum (the number of fax machines that person will ever send a fax to or receive a fax from)...

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